[R-sig-ME] Question about misspecified multilevel model

[Christopher David Desjardins]

Hi,

I have a question that's potentially off-topic but that I'm hoping that
someone here can shed some insight on.

Assume that I know that I know my true model and that my true is a
three-level model. My observations are such that I have a measurement on a
student nested within a classroom nested within a school. The true model
would be:

Y_ijk = pi_0jk + e_ijk  # student within classroom within schools (1st
level)

pi_0jk = beta_j0k + r_p0k  # classroom within schools (2nd level)

beta_j0k = gamma_pq0 + u_pqk  # schools (3rd level)

The model in lmer would be:
classroom <- read.csv("http://www-personal.umich.edu/~bwest/classroom.csv")
library("lme4")
correct.mod <- lmer(mathgain ~ (1 | schoolid/classid), data = classroom)

What I am wondering about is, if I were to omit that second level, the
whole classroom within schools equation, where would that variance that
would end up as the random intercept go? Would it go to the random
intercept for school or would it go down to the residual?  The model I am
referring to is below:

misspecified.mod <- lmer(mathgain ~ (1 | schoolid), data = classroom)
summary(correct.mod); summary(misspecified.mod)

It looks like the variances for both the residual and the random intercept
for school change. But maybe they do in a predictable way?

If someone could suggest a paper has an answer or better that would be very
helpful.

Chris

	[[alternative HTML version deleted]]